This abstract algebra textbook takes an integrated approach that highlights the similarities of fundamental algebraic structures among a number of topics. The book begins by introducing groups, rings, vector spaces, and fields, emphasizing examples, definitions, homomorphisms, and proofs. The goal is to explain how all of the constructions fit into an axiomatic framework and to emphasize the importance of studying those maps that preserve the underlying algebraic structure. This fast-paced introduction is followed by chapters in which each of the four main topics is revisited and deeper results are proven.
The second half of the book contains material of a more advanced nature. It includes a thorough development of Galois theory, a chapter on modules, and short surveys of additional algebraic topics designed to whet the reader's appetite for further study.
This book is intended for a first introduction to abstract algebra and requires only a course in linear algebra as a prerequisite. The more advanced material could be used in an introductory graduate-level course.
Joseph H. Silverman: Brown University, Providence, RI
Preface
Chapter 1. A Potpourri of Preliminary Topics 1.1. What Are Definitions, Axioms, and Proofs? 1.2. Mathematical Credos to Live By! 1.3. A Smidgeon of Mathematical Logic and Some Proof Techniques 1.4. A Smidgeon of Set Theory 1.5. Functions 1.6. Equivalence Relations 1.7. Mathematical Induction 1.8. A Smidgeon of Number Theory 1.9. A Smidgeon of Combinatorics Exercises
Chapter 2. Groups — Part 1 2.1. Introduction to Groups 2.2. Abstract Groups 2.3. Interesting Examples of Groups 2.4. Group Homomorphisms 2.5. Subgroups, Cosets, and Lagrange's Theorem 2.6. Products of Groups Exercises
Chapter 3. Rings — Part 1 3.1. Introduction to Rings 3.2. Abstract Rings and Ring Homomorphisms 3.3. Interesting Examples of Rings 3.4. Some Important Special Types of Rings 3.5. Unit Groups and Product Rings 3.6. Ideals and Quotient Rings 3.7. Prime Ideals and Maximal Ideals Exercises
Chapter 4. Vector Spaces — Part 1 4.1. Introduction to Vector Spaces 4.2. Vector Spaces and Linear Transformations 4.3. Interesting Examples of Vector Spaces 4.4. Bases and Dimension Exercises
Chapter 5. Fields — Part 1 5.1. Introduction to Fields 5.2. Abstract Fields and Homomorphisms 5.3. Interesting Examples of Fields 5.4. Subfields and Extension Fields 5.5. Polynomial Rings 5.6. Building Extension Fields 5.7. Finite Fields Exercises
Chapter 6. Groups — Part 2 6.1. Normal Subgroups and Quotient Groups 6.2. Groups Acting on Sets 6.3. The Orbit-Stabilizer Counting Theorem 6.4. Sylow's Theorem 6.5. Two Counting Lemmas 6.6. Double Cosets and Sylow's Theorem Exercises
Chapter 7. Rings — Part 2 7.1. Irreducible Elements and Unique Factorization Domains 7.2. Euclidean Domains and Principal Ideal Domains 7.3. Factorization in Principal Ideal Domains 7.4. The Chinese Remainder Theorem 7.5. Field of Fractions 7.6. Multivariate and Symmetric Polynomials Exercises
Chapter 8. Fields — Part 2 8.1. Algebraic Numbers and Transcendental Numbers 8.2. Polynomial Roots and Multiplicative Subgroups 8.3. Splitting Fields, Separability, and Irreducibility 8.4. Finite Fields Revisited 8.5. Gauss's Lemma and Eisenstein's Irreducibility Criterion 8.6. Ruler and Compass Constructions Exercises
Chapter 9. Galois Theory: Fields+Groups 9.1. What Is Galois Theory? 9.2. A Quick Review of Polynomials and Field Extensions 9.3. Fields of Algebraic Numbers 9.4. Algebraically Closed Fields 9.5. Automorphisms of Fields 9.6. Splitting Fields — Part 1 9.7. Splitting Fields — Part 2 9.8. The Primitive Element Theorem 9.9. Galois Extensions 9.10. The Fundamental Theorem of Galois Theory 9.11. Application: The Fundamental Theorem of Algebra 9.12. Galois Theory of Finite Fields 9.13. A Plethora of Galois Equivalences 9.14. Cyclotomic Fields and Kummer Fields 9.15. Application: Insolubility of Polynomial Equations by Radicals 9.16. Linear Independence of Field Automorphisms Exercises
Chapter 10. Vector Spaces — Part 2 10.1. Vector Space Homomorphisms (aka Linear Transformations) 10.2. Endomorphisms and Automorphisms 10.3. Linear Transformations and Matrices 10.4. Subspaces and Quotient Spaces 10.5. Eigenvalues and Eigenvectors 10.6. Determinants 10.7. Determinants, Eigenvalues, and Characteristic Polynomials 10.8. Inifinite-Dimensional Vector Spaces Exercises
Chapter 11. Modules — Part 1:Rings+Vector-Like Spaces 11.1. What Is a Module? 11.2. Examples of Modules 11.3. Submodules and Quotient Modules 11.4. Free Modules and Finitely Generated Modules 11.5. Homomorphisms, Endomorphisms, Matrices 11.6. Noetherian Rings and Modules 11.7. Matrices with Entries in a Euclidean Domain 11.8. Finitely Generated Modules over Euclidean Domains 11.9. Applications of the Structure Theorem Exercises
Chapter 12. Groups — Part 3 12.1. Permutation Groups 12.2. Cayley's Theorem 12.3. Simple Groups 12.4. Composition Series 12.5. Automorphism Groups 12.6. Semidirect Products of Groups 12.7. The Structure of Finite Abelian Groups Exercises
Chapter 13. Modules — Part 2: Multilinear Algebra 13.1. Multilinear Maps and Multilinear Forms 13.2. Symmetric and Alternating Forms 13.3. Alternating Forms on Free Modules 13.4. The Determinant Map Exercises
Chapter 14. Additional Topics in Brief 14.1. Sets Countable and Uncountable 14.2. The Axiom of Choice 14.3. Tensor Products and Multilinear Algebra 14.4. Commutative Algebra 14.5. Category Theory 14.6. Graph Theory 14.7. Representation Theory 14.8. Elliptic Curves 14.9. Algebraic Number Theory 14.10. Algebraic Geometry 14.11. Euclidean Lattices 14.12. Non-Commutative Rings 14.13. Mathematical Cryptography Exercises
Sample Syllabi List of Notation List of Figures Index